You're going to have a hard time extending / interfacing with that doorbell/phone flasher. Does that one make a sound too (i'm guessing no)?
If the house has an old hardwired doorbell and chime, there's a product that can detect the button press and give you a digital output you can feed into something else. The ELK-930 includes 2 doorbell detector circuits and one telephone ring detector, but for it's simplicity it's a bit overpriced.
Newer houses typically have smoke detectors that are powered off of 120VAC. If you have this type of interconnected smoke detector, I'd add 1 or 2 hard wired smoke detectors with strobe lights. Interconnected carbon monoxide detectors would also work with these. The strobe may not be enough to wake you from a deep sleep. Regarding the clarity product you linked, from a quick look I think Serene Innovations CentralAlert system is better.
I do like the reliability of hardwired, but there are more products like visual doorbells than there used to be (though most are wireless). Most aren't designed to integrate with other things or the internet, how else are they going to force you to buy a bunch of
their sensors, buttons, and displays...
as only china can...
https://www.amazon.com/dp/B071NGS5MF/
https://www.amazon.com/Honeywell-RDWL515A2000-Portable-Wireless-Doorbell/dp/B01HNSFTBQ/ look at different styles in drop down
https://www.amazon.com/Honeywell-RCWL330A1000-P4-Premium-Portable-Wireless/dp/B001G0MAVA/
more details and models on their website
Doorbells | Honeywell
*** Looks like some of Honeywell's products use the European 868MHz band. Models listed on the 'yourhome' site
should be US models, but idk if they actually use a different frequency. I'm not sure if you can make different buttons and sensors light up different colors or not. If not they really didn't understand the market for these.
phone flashers are a dime a dozen and can be plugged in anywhere there's a phone jack, you just have to watch the total REN and they generally only alert to phone calls. You can also tie ring detection into some other system. Do you use captel, tty, or a video relay service most often? Is it POTS, VoIP or other?
A smart watch that can vibrate could be a good way to get your attention. I wonder if there are any low frequency subwoofer/bass shaker type products that could make a sound you'd notice/feel without being too loud or obnoxious. I know there are products like bed vibrators/shakers for alarm clocks.
There are a lot of cheap android tablets and prepaid phones. Consider expanding the number of devices that get the alerts. There's a thread on wall mounting a fire tablet. You can make use of things like the device's flash or use a custom home automation display that flashes between color or changes color.
The most versatile setup would be more of a custom automation / alarm system. An ELK alarm system for example could be extremely powerful while being simpler in some ways than other home automation systems. You can hook the output from an alarm or automation system to strobe lights, sirens, and bed shakers pretty easily. Some alarm systems are more flexible in what you can make them do. Interfacing with things that can change color is possible too.
A few interesting products:
Z-Wave Plus RGB Smart Bulb (Works with Alexa & Google Home with Hub) - Monoprice.com
https://www.amazon.com/Fibaro-Micro-Controller-Z-wave-Strips/dp/B00P1N68FW/
https://www.amazon.com/EZMultiPli-Z-Wave-Multi-Sensor-Plug-Temperature/dp/B00Q3OR0CI/
https://www.amazon.com/HomeSeer-HS-WD200-Z-Wave-Scene-Capable-Dimmer/dp/B079F38TPF/
Lighting – Tagged "Special Features_RGB LED Indicators" – HomeSeer Store
TRÅDFRI Dimming kit - IKEA not zwave